Question

Tell whether you should use an open dot or a closed dot on the graph of the inequality. \(x \geq 5\)

Short answer

A closed dot should be used on the graph of the inequality. The closed dot represents that the value of x is 'greater than or equal to' 5.

Step-by-step solution

Understand the inequality

Analyze the inequality, this is \(x \geq 5\). It can be read as 'x is greater than or equal to 5'.

Determine the correct dot to use

The inequality involves 'greater than or equal to'. In graphical representation, this is represented by a closed dot, as it includes the 'equal to' part.

Graphical Representation

Draw a number line and place a closed dot on the number 5. The line extends to the right because this signifies 'greater than'.

Key concepts

Closed Dot Representation

In graphing inequalities, the type of dot we use on the number line holds a special significance, as it tells us whether the number at that point is included in the solution set. A closed dot, which is a filled-in circle, is used when a particular number is part of the solution.

For instance, when dealing with the inequality \(x \geq 5\), we use a closed dot on the number 5. This visually conveys that 5 is included in the set of numbers that satisfy the inequality, along with all numbers greater than 5. Using the correct dot is crucial as it ensures the accurate representation of solutions on the graph.

Number Line

The number line is a fundamental tool in understanding and visualizing mathematical concepts, especially when it comes to inequalities. It is a straight, horizontal line that has points which correspond to numbers. The placement of these numbers is in ascending order from left to right.

When graphing an inequality like \(x \geq 5\), we use the number line to first position the number 5 on it. As we determined a closed dot is appropriate, we place the closed dot on 5. Then, we draw a ray extending to the right from the closed dot to represent all the numbers that are greater than 5. This helps learners to not only visualize the range of solutions but to understand the concept of infinity in such inequalities as well.

Inequality Symbols

Inequality symbols are mathematical symbols that express the relative size or order of two values. The most common symbols include 'less than' (\(<\)), 'greater than' (\(>\)), 'less than or equal to' (\(\leq\)), and 'greater than or equal to' (\(\geq\)).

For the inequality \(x \geq 5\), the symbol \(\geq\) tells us that the value of \(x\) is either greater than or exactly equal to 5. Reading and interpreting these symbols correctly is vital for students to solve and graph inequalities accurately. Remember that the direction of the symbol points toward the smaller number and opens up to the larger number, thus giving a visual cue for comparison.